Solve for $x$ : $ 2|x - 1| - 10 = 1|x - 1| + 7 $
Answer: Subtract $ {1|x - 1|} $ from both sides: $ \begin{eqnarray} 2|x - 1| - 10 &=& 1|x - 1| + 7 \\ \\ { - 1|x - 1|} && { - 1|x - 1|} \\ \\ 1|x - 1| - 10 &=& 7 \end{eqnarray} $ Add ${10}$ to both sides: $ \begin{eqnarray} 1|x - 1| - 10 &=& 7 \\ \\ { + 10} &=& { + 10} \\ \\ 1|x - 1| &=& 17 \end{eqnarray} $ Simplify: $ |x - 1| = 17$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 1 = -17 $ or $ x - 1 = 17 $ Solve for the solution where $x - 1$ is negative: $ x - 1 = -17 $ Add ${1}$ to both sides: $ \begin{eqnarray} x - 1 &=& -17 \\ \\ {+ 1} && {+ 1} \\ \\ x &=& -17 + 1 \end{eqnarray} $ $ x = -16 $ Then calculate the solution where $x - 1$ is positive: $ x - 1 = 17 $ Add ${1}$ to both sides: $ \begin{eqnarray} x - 1 &=& 17 \\ \\ {+ 1} && {+ 1} \\ \\ x &=& 17 + 1 \end{eqnarray} $ $ x = 18 $ Thus, the correct answer is $x = -16 $ or $x = 18 $.